A horizontal line charge of length L has a uniform charge density + lambda Consi

Question

A horizontal line charge of length L has a uniform charge density + lambda Consider horizontal to be the x-axis, with +x being to the right. Take x=0 to be on this axis, 5 meters to the right of the line charge. Which of these integrals correctly expresses the magnitude of the electric field at a point which is 8 meters to the left of the line charge? (a) integral_-5^-(5+L) k lambda dx/(8 + L + x)^2 (b) integral_-5^-5 + 2 k lambda dx/(13 + L + x)^2 (c) integral_-5^-5 + 6 k lambda dx/(k + L + x)^2 (d) integral_-5^-(5 + L) k lambda dx/(52 + L + x)^2

Explanation / Answer

the right end of the line charge is at x=-5 m

the left end is at x=-5-L

so the limits of integration is from x=-5 to -5-L

consider a small length dx on the line charge

at coordinate x.

then charge on this small segment=lambda*dx

a point which is 8 meters to the left of line charge,

will be at adistance of 8+L+5+x=x+L+13 from the small segment dx.

so electric field =dE=k*lambda*dx/(x+L+13)^2

hence option d is correct.

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